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Theorem lvoli3 34590
Description: Condition implying a 3-dim lattice volume. (Contributed by NM, 2-Aug-2012.)
Hypotheses
Ref Expression
lvoli3.l  |-  .<_  =  ( le `  K )
lvoli3.j  |-  .\/  =  ( join `  K )
lvoli3.a  |-  A  =  ( Atoms `  K )
lvoli3.p  |-  P  =  ( LPlanes `  K )
lvoli3.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
lvoli3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  e.  V )

Proof of Theorem lvoli3
Dummy variables  y 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 1000 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  X  e.  P
)
2 simpl3 1001 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  Q  e.  A
)
3 simpr 461 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  -.  Q  .<_  X )
4 eqidd 2468 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  =  ( X 
.\/  Q ) )
5 breq2 4451 . . . . . 6  |-  ( y  =  X  ->  (
r  .<_  y  <->  r  .<_  X ) )
65notbid 294 . . . . 5  |-  ( y  =  X  ->  ( -.  r  .<_  y  <->  -.  r  .<_  X ) )
7 oveq1 6292 . . . . . 6  |-  ( y  =  X  ->  (
y  .\/  r )  =  ( X  .\/  r ) )
87eqeq2d 2481 . . . . 5  |-  ( y  =  X  ->  (
( X  .\/  Q
)  =  ( y 
.\/  r )  <->  ( X  .\/  Q )  =  ( X  .\/  r ) ) )
96, 8anbi12d 710 . . . 4  |-  ( y  =  X  ->  (
( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y 
.\/  r ) )  <-> 
( -.  r  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  r ) ) ) )
10 breq1 4450 . . . . . 6  |-  ( r  =  Q  ->  (
r  .<_  X  <->  Q  .<_  X ) )
1110notbid 294 . . . . 5  |-  ( r  =  Q  ->  ( -.  r  .<_  X  <->  -.  Q  .<_  X ) )
12 oveq2 6293 . . . . . 6  |-  ( r  =  Q  ->  ( X  .\/  r )  =  ( X  .\/  Q
) )
1312eqeq2d 2481 . . . . 5  |-  ( r  =  Q  ->  (
( X  .\/  Q
)  =  ( X 
.\/  r )  <->  ( X  .\/  Q )  =  ( X  .\/  Q ) ) )
1411, 13anbi12d 710 . . . 4  |-  ( r  =  Q  ->  (
( -.  r  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  r ) )  <-> 
( -.  Q  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  Q ) ) ) )
159, 14rspc2ev 3225 . . 3  |-  ( ( X  e.  P  /\  Q  e.  A  /\  ( -.  Q  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  Q ) ) )  ->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y  .\/  r
) ) )
161, 2, 3, 4, 15syl112anc 1232 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y 
.\/  r ) ) )
17 simpl1 999 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  K  e.  HL )
18 hllat 34377 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
1917, 18syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  K  e.  Lat )
20 eqid 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
21 lvoli3.p . . . . . 6  |-  P  =  ( LPlanes `  K )
2220, 21lplnbase 34547 . . . . 5  |-  ( X  e.  P  ->  X  e.  ( Base `  K
) )
231, 22syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  X  e.  (
Base `  K )
)
24 lvoli3.a . . . . . 6  |-  A  =  ( Atoms `  K )
2520, 24atbase 34303 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
262, 25syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  Q  e.  (
Base `  K )
)
27 lvoli3.j . . . . 5  |-  .\/  =  ( join `  K )
2820, 27latjcl 15541 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( X  .\/  Q )  e.  ( Base `  K
) )
2919, 23, 26, 28syl3anc 1228 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  e.  ( Base `  K ) )
30 lvoli3.l . . . 4  |-  .<_  =  ( le `  K )
31 lvoli3.v . . . 4  |-  V  =  ( LVols `  K )
3220, 30, 27, 24, 21, 31islvol3 34589 . . 3  |-  ( ( K  e.  HL  /\  ( X  .\/  Q )  e.  ( Base `  K
) )  ->  (
( X  .\/  Q
)  e.  V  <->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y  .\/  r
) ) ) )
3317, 29, 32syl2anc 661 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( ( X 
.\/  Q )  e.  V  <->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y 
.\/  r ) ) ) )
3416, 33mpbird 232 1  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  e.  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434   Latclat 15535   Atomscatm 34277   HLchlt 34364   LPlanesclpl 34505   LVolsclvol 34506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-lplanes 34512  df-lvols 34513
This theorem is referenced by:  dalem9  34685  dalem39  34724
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